Etymology
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red cent (n.)

type of copper penny, 1839, American English, from red (adj.1) + cent. Pure copper pennies were issued 1793–1857, then replaced by ones of copper-nickel and, after 1864, bronze. The old cents were disused, but the phrase remained colloquial as a mere emphatic of cent, usually in the negative (don't have a ... not worth a ...). "Red" has been the color of copper, brass, and gold since ancient times.

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Definitions of red cent

red cent (n.)
something of little value;
not worth one red cent
Synonyms: damn / darn / hoot / shit / shucks / tinker's damn / tinker's dam
From wordnet.princeton.edu